How Many Different Bowls of Ice Cream Can Be Made?
An ice-cream shop sells 12 different flavors, including mango and lemon. For a bowl, a customer can choose 4 flavors, with the possibility of choosing the same flavor more than once. We explore various combinatorial scenarios to determine the number of different bowls one can make.
Introduction to Stars and Bars Combinatorics
Stars and bars combinatorics is a powerful tool for solving problems involving the distribution of indistinguishable objects into distinct groups. The theorem states that for any pair of positive integers (n) and (k), the number of (k)-tuples of non-negative integers whose sum is (n) is equal to the number of multisets of cardinality (k) taken from a set of size (n-1). This number is given by the binomial coefficient (binom{n k-1}{k-1}).
Scenario A: How Many Different Bowls Can Be Made?
In this scenario, we need to determine the number of ways to create a bowl of ice cream by choosing 4 flavors from 12 different options, where flavors can be repeated. Here, (n 4) and (k 12), meaning we have 4 slots to fill and 12 different flavors available for selection.
According to the stars and bars theorem, the number of solutions is given by:
[binom{4 12-1}{12-1} binom{15}{11} 1365]Therefore, there are 1,365 different ways to create a bowl of ice cream with 4 flavors chosen from 12 options.
Scenario B: How Many Different Bowls Contain Both Lemon and Mango?
If we want to find the number of bowls that include both lemon and mango, we use a similar approach. Here, (n 2) (since only 2 flavors need to be selected) and (k 12). However, lemon and mango must be included, so effectively, we have 2 flavors already chosen and 2 slots left to fill with any of the 12 flavors.
We reduce the problem to (n 2) and (k 10) (because 2 flavors are fixed and 2 flavors are chosen from the remaining 10), giving us:
[binom{2 10-1}{10-1} binom{11}{9} 55]However, we need to account for the fact that lemon and mango are already chosen. Thus, the problem simplifies to choosing the remaining 2 flavors from the remaining 10 flavors, which is:
[binom{2 8-1}{8-1} binom{9}{7} 36]Or more simply, using the combination directly:
[binom{13}{11} 78]Therefore, there are 78 different ways to create a bowl containing both lemon and mango.
Scenario C: How Many Bowls Don’t Contain Lemon but Contain Mango?
In this scenario, we want to find the number of bowls that contain mango but not lemon. This means we have 11 flavors to choose from and need to choose 4 flavors, with mango fixed as one of them. Thus, we have 3 flavors to choose from the remaining 11 options and 3 slots to fill.
The problem is reduced to (n 3) and (k 11), giving us:
[binom{3 11-1}{11-1} binom{13}{10} 286]Therefore, there are 286 different ways to create a bowl that contains mango but not lemon.
Conclusion
By using combinatorial methods, specifically stars and bars, we have determined the different ways to create bowls of ice cream based on the given constraints. The scenarios demonstrated demonstrate the power of this theorem in solving real-world combinatorial problems.
Key terms: combinatorics, ice cream combinations, stars and bars.